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Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

January 29, 2007

itex2MML 1.1.9

Note that, because of this bug, the spacing on all of the array-like environments is a bit screwed-up in Mozilla. If Roger ever gets around to fixing that bug, the spacing will be much improved, and I’ll be able to fine-tune it to look really sharp.

Update (2/7/2007):

Per Lieven le Bruyn’s suggestion, the MacOSX binary of itex2MML is now a Universal Binary.

January 27, 2007

Barbarians at the PAC

I was a student at Harvard, when the first recording of Einstein on the Beach was released. Philip Glass did a record signing at the Harvard COOP, and I decided to go down to check it out. Naturally, they played the opera on the stereo, as the composer sat at a table, signing records and chatting amiably with the customers. After about an hour of this, one of the COOP staffers cracked. Shaking visibly, she strode over to the stereo, said “I just can’t take this any more!” and — right in front of the composer — took Einstein on the Beachoff the stereo and put something else on in its stead.

Everyone looked slightly embarrassed, at what had just transpired. Except for Philip Glass, who kept right on chatting and signing records, as if nothing out-of-the-ordinary had taken place.

It was around this time that I read J.M. Coetzee’s novella, Waiting for the Barbarians. It was, or so it appeared, an allegory for the brutalities of South African apartheid regime’s battle with the black nationalists.

The libretto of the Philip Glass opera, whose American premier was performed, here, by the Austin Lyric Opera is quite faithful to Coetzee’s novella. The set is stunning, and the music sweeps us along with the Magistrate, and his town’s, descent into the abyss.

The barbarians, whose threat, Colonel Joll assures the Magistrate, requires the “strictly temporary” suspension of various civil liberties (and the rather graphic torture of as many “prisoners” as he can scoop up) seem much closer to home than they did back then, not so easily dismissed as an allegory about the troubles in some faraway land. And this, too, lends a certain urgency to the unfolding drama.

If there are still tickets to the final performance, Monday night, … go.

January 25, 2007

Gravitational Leptogenesis?

I was very excited when I saw a recent paper by Alexander, Peskin and Sheikh-Jabbari (which, I guess, is based on this older paper). They claimed to produce an acceptable lepton-asymmetry during inflation if the inflaton is a (CP-odd) pseudoscalar field, ϕ\phi, with a coupling

(1)F(ϕ)RR˜
F(\phi) R\tilde{R}

to the curvature, where F(ϕ)F(\phi) is an odd function of ϕ\phi.

Fluctuations generated during inflation drive an expectation-value for RR˜R\tilde{R}, which generates a lepton-asymmetry via the anomaly,

(2)∂μjLμ=N16π2RR˜
\partial_\mu j^\mu_L = \frac{N}{16\pi^2} R\tilde{R}

For reasons, that will become apparent shortly, however, they proceed to say something very unconventional about the anomaly coefficient, NN:

In general, when heavy right-handed neutrinos are also added to the Standard Model, as is done in the seesaw mechanism for explaining the smallness of the neutrino mass, (2) will be correct in an effective theory valid below a scale μ\mu, of order of the right-handed neutrino mass. More concretely, NN can in general be a function of energy. At low energies, below the right-handed neutrino mass scale N=3N=3. At higher energies, NN could be anywhere between zero to three, depending on the details of the particle physics invoked.

Umh … no. That’s not how anomalies work. The anomaly depends only on the massless spectrum and the anomaly coefficient does not run with energy.

Now, it’s true that, above the scale μ\mu, one can treat the right-handed neutrino, Ψ\Psi, as massless. In that theory, there’s a new current which is non-anomalous.
∂μ(jLμ+jΨμ)=0
\partial_\mu (j^\mu_L + j^\mu_\Psi) = 0
However, once one includes the Majorana mass for Ψ\Psi, jΨμj^\mu_\Psi is not conserved, even classically. The correct, classically-conserved, current in the theory which includes the mass term, μ\mu is jLμj^\mu_L. And the anomaly in that current is the same, whether one calculates it in the high- or low-energy theory.

The motivation for the seemingly bizarre statement about the anomaly (2) becomes apparent a little later on. One would naïvely expect the gravitational contribution to the lepton asymmetry, during inflation, to be tiny, suppressed by at least a factor of (HMpl)2{\left(\tfrac{H}{M_{\text{pl}}}\right)}^2. They, however, claim that there’s an enhancement over the naïve answer by a factor of (μH)6{\left(\tfrac{\mu}{H}\right)}^6. This enhancement comes about because, in their calculation, ⟨RR˜⟩\langle R\tilde{R}\rangle is UV-divergent, dominated by short-distance effects near the cutoff, μ\mu.

I don’t really understand their calculation. But, if they are correct that, in the effective theory, ⟨RR˜⟩∼μ6\langle R\tilde{R}\rangle \sim \mu^6, then what this means is that, in the full theory, where one includes loops of Ψ\Psi, the contribution of those extra diagrams cancels the erstwhile UV divergence, yielding a finite (but μ\mu-dependent) answer. I don’t see why there should be such a cancellation, here.

January 24, 2007

Tech Week

This year, again, my daughter is in a musical at the Zach Scott Theater. They open Saturday, so this week is Tech Week, with 3 1/2 hour rehearsals every night. Which has made made our lives kinda hectic, as of late.

Things will settle down, a bit, once the show opens.

Tonight was their first full dress rehearsal, in front of an audience. I was home, but my wife said it went well. From the bits I’ve seen, it promises to be a terrific show. So, if you’re in Austin, and have a Saturday afternoon free …

Part I: The world is full of stupid people.Part II: Too many stupid people go to college.Part III: We should spend more money on the tiny fraction of people who are smart.

As Chad notes, Murray covers everything he touches with a thick coating of slime. Which makes one feel particularly icky when he, occasionally, does touch on something halfway sensible (the defects of No Child Left Behind, the scanting of gifted education). But, I guess, my ickiest moment came in reading the lede to part III

If “intellectually gifted” is defined to mean people who can become theoretical physicists, then we’re talking about no more than a few people per thousand and perhaps many fewer. They are cognitive curiosities, too rare to have that much impact on the functioning of society from day to day.

Couldn’t he, please, have chosen someone else to place at the top of his intellectual heap? AEI Fellows, perhaps?

January 17, 2007

One Thing Led to Another …

Update (2/15/2007):

I was entering some of the notes, into the wiki, of the discussions with my collaborators, when Instiki more-or-less ground to a halt. Turns out that Ruby is really slow at string-manipulation and Regexp-based BlueCloth just can’t handle any remotely complicated input. I wrote a rant about it and a commenter suggested Maruku as an alternative Markdown implementation.

January 12, 2007

MTValidate 0.5

In my quest to improve the user experience hereabouts, I decided to convert this MovableType installation to FastCGI. I’ll explain what’s involved in that in another post, but I did run into a nasty stumbling block: the MTValidate plugin, that we rely on heavily, was incompatible with FastCGI.

It was not (at first) apparent why MTValidate was failing in a FastCGI environment; there were lots of crufty bits of W3C Validator code, which I thought might be the culprits. So I engaged in a lot of code clean-up until I eventually figured out that the problem was its use of IPC::Open3, which is not compatible with FastCGI.

Once I knew what the problem was, it was easy to fix (as written, there was no good reason for the W3C Validator code to be using IPC::Open3 in the first place). But, along the way, I found that I had

January 5, 2007

Full Disclosure

A little over a month ago, Mike Stay noticed something really bad. Our MovableType installation seemed to be vulnerable to script injection. In fact, it did not seem to be doing any sanitization on user input at all! You could type any (valid) XHTML input a comment, and it would be accepted.

This was very puzzling, as comments used to be sanitized. I contacted Zack Ajmal, who has an MT installation similar to mine, which also seemed to be vulnerable. Searching around the ‘Net, some MT 3.3 blogs seemed to be vulnerable; others did not. Eventually, it transpired that those who had disabled the “nofollow” plugin , had inadvertently disabled comment-sanitization as well1. SixApart, in their wisdom, had bypassed the sanitization code in the body of the application (where it used to take place) and did the sanitization in the plugin instead. Which meant that if you disabled the plugin …

This was bad, and Zack contacted SixApart on December 6. They asked him to hold off on disclosing the vulnerability for 30 days. That interval has passed, and Zack has published details of the vulnerability on his blog. I’m doing so, too, in the interest of getting the word out to those who may be vulnerable.

Until SixApart issues an update, you have two options for fixing the problem

Re-enable the “nofollow” plugin.

Add an explicit sanitize="1" attribute to any MovableType tag that needs sanitizing:

January 2, 2007

Choptuik and the Pomeron

One of the most intriguing papers of 2006 was by Álvarez-Gaumé,Gómez and Vázquez-Mozo. They noticed a remarkable numerical coincidence between the Choptuik exponent for the formation of 5D blackholes, and a certain critical exponent associated to the BFKL Pomeron in 4D gauge theories.

The latter emerges in perturbative gauge theories, in the regime s≫−t≫ΛQCD2s\gg -t \gg \Lambda_{\text{QCD}}^2, from resumming a set of diagrams to all orders in (αslog(s/t))n{\left(\alpha_s \log(s/t)\right)}^n, n≥0n\geq 0.

For t∼0t\sim 0, the scattering amplitude of two hadrons can be written in terms of impact functions, Φi(k⇀i2)\Phi_i(\vec{k}_i^2), which are functions of the 2D transverse momenta, k⇀i\vec{k}_i, and a kernel, K(k⇀12,k⇀22,s)K(\vec{k}_1^2,\vec{k}_2^2,s)

where we’ve defined the dimensionless parameters
y=g2N(2π)2log(sμ2),τi=log(k⇀i2μ2)
y = \frac{g^2 N}{(2\pi)^2}\log\left(\frac{s}{\mu^2}\right),\quad \tau_i = \log\left(\frac{\vec{k}_i^2}{\mu^2}\right)
with μ\mu some characteristic energy scale for the process, and the function χ0\chi_0 is written in terms of digamma functions
χ0(η)=2ψ(1)−ψ(η)−ψ(1−η)
\chi_0(\eta) =2\psi(1)-\psi(\eta)-\psi(1-\eta)

The BFKL amplitude rises faster than slog2(s)s\log^2(s), and eventually violates the unitarity bound. The idea is to define a critical exponent by evaluating the above integral in a saddle point approximation, with the saddle point located at η=η*\eta=\eta_*, given by

(3)η*χ′0(η*)=χ0(η*)
\eta_* \chi'_0(\eta_*)=\chi_0(\eta_*)

One then finds the value of yy, as a function of τ1−τ2\tau_1-\tau_2, which saturates the unitarity bound.

They would like to identify this critical exponent with Choptuik exponent

(6)MBH∝(p−p*)γ
M_{\text{BH}} \propto (p-p_*)^\gamma

for 5D gravitational collapse. The space of initial value data for the gravitational collapse has two basins of attraction1, corresponding to empty space and to the formation of a blackhole. These two basins of attraction are separated by a critical surface of codimension-1. By tuning a single parameter (pp) in the initial value data, one can tune to this critical surface. For p>p*p\gt p_*, a blackhole forms, and its mass is universally determined by this scaling exponent, γ\gamma. Numerically, in 5D, γ=0.408±2%\gamma=0.408\pm2\%.

Why should these two exponents be related?

Álvarez-Gaumé et al give some arguments, but I don’t find them terribly convincing.

The next-to-leading-order corrections to the BFKL kernel are large. In a conformally-invariant theory, they are not as large as in QCD, but they are still substantial enough to make one wonder whether the apparent agreement between γBFKL\gamma_{\text{BFKL}} and γ\gamma is spurious.

Leaving that aside, one wonders why a weak-coupling calculation of the BFKL exponent has anything to do with a computation in classical 5D GR. Weak coupling in the gauge theory corresponds to a small radius, highly-curved AdS space, where α′\alpha' corrections to GR are substantial. Now, it’s true (as they argue) that blackhole creation at threshold involves regions of large curvature (where the corrections to GR are, presumably, substantial). But the whole interest of Choptuik’s result is that it is supposed to be robust against such corrections. The scaling behaviour, (6), for instance, continues to hold reasonably far from the critical surface, where the curvature at the apparent horizon is small enough that GR is trustworthy.

I find it very plausible that there’s a relation between the bulk gravitational collapse problem and Pomeron physics. But I don’t (yet) see how the particulars of the leading-log BFKL calculation connects to the physics of Choptuik scaling. I hope 2007 brings some insight into whether their observation is a mere coincidence, or the first hint of something very deep.

1 In many “realistic” systems, there’s a third basin of attraction, corresponding to the formation of stable stars.

January 1, 2007

TeX Macros

Instiki provides a (broken, but it’s there) output to LaTeX option. Now that we’re producing a significant amount of interesting XHTML+MathML content, there’s general interest in being able to reuse that content in (La)TeX. A lot of effort has gone into converters from TeX-like input into XHTML+MathML. Not so much has gone into converters which go the other way.